A quantum algebra approach to discrete equations on uniform lattices

A quantum algebra method for deducing the symmetries of discrete equations on uniform lattices is proposed. In principle, such a procedure can be appl ied to discretizations in a single coordinate (space or time) and the symme tries obtained in this may are indeed differential-difference operators. Fi rstly, the method is illustrated on two known examples that have been also analysed from the usual Lie symmetry approach: a uniform space lattice disc retization of the (1 + 1) free heat-Schrodinger equation associated to a qu antum Schrodinger algebra, and a discrete space (1 + 1) wave equation provi ded by a quantum so(2, 2) algebra. Furthermore, we construct a discrete spa ce (2 + 1) wave equation from a new quantum so(3, 2) algebra, to show that this method is useful in higher dimensions. Time discretizations are also c ommented.